Recall that a \(k\) -toast in a Borel graph \(G\)  is a Borel collection \(\mathcal {T}\)  of finite connected subsets of \(V(G)\)  such that \(\bigcup _{K\in \mathcal {T}}E(K)=E(G)\) , and for every pair \(K, L \in \mathcal {T}\)  either \((N^k(K)\cup K) \cap L = \emptyset \)  or \(K \cup N^k(K) \subseteq L\) , or \(L \cup N^k(L) \subseteq K\) . By the result of Gao, Jackson, Krohne, and Seward (see [1]}), any free Borel \(\mathbb {Z}^d\)  action admits a \(k\) -toast \(\mathcal {T}\)  such that the boundary of each \(K\in \mathcal {T}\)  is visible from infinity (i.e., \(K\)  has no holes). Note that \(\mathcal {T}\)  satisfies Properties (1) and (2) of Definition REF ,
and we claim that it also satisfies Property (3) as long as \(k>2.\)  To see this, we will use a result of Deuschel and Pisztora [2]} which says that if a finite, connected set \(H\subset \mathbb {Z}^d\)  has no holes, then \(N^2(H)\setminus H\)  is connected. Thus,
since each \(K\in \mathcal {T}\)  has no holes, the set \(N^2(K)\setminus K\)  is connected, and so for every \(L \in \mathcal {T}\)  the induced subgraph on \(L \setminus \bigcup _{L \supsetneq K \in \mathcal {T}} K\)  is connected, as any path through smaller tiles \(K\)  can be rerouted through \(N^2(K)\setminus K\) .
